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Consider an auto-regressive process $x[n]=0.6x[n-1]+g[n]$ where $g[n]$ is unit variance white noise, Now if I am finding the Autocorrelation matrix $R_{x}(k)$, I am encountering a basic problem.

$R_{x}(k)=E\{x[n]x[n-k]\}=E\{(0.6x[n-1]+g[n])(x[n-k])\}=0.6R_{x}(k-1)+\delta(k)$

however for the step $E\{(0.6x[n-1]+g[n])(x[n-k])\}$ if I am expanding $x[n-k]$ inside the expectation, I am getting $E\{(0.6x[n-1]+g[n])(0.6x[n-1-k]+g[n])\}=0.36R_{x}[k]+\delta(k)$. Where I am I going wrong?

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The mistake is that $R_x(k)$ is defined for fixed $n$. Hence, the only recursion which makes sense to consider is $$R_{x}(k)=E\{x[n]x[n-k]\}=E\{x[n](.6x[n-k-1]+g[n-k])\}=.6R_{x}(k+1)+E\{x[n]g[n-k]\}.$$

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    Thanks , but why is that expanded for $x[n-k]$, it should be kept as it is and therm $x[n]$ should be expanded right? – 2017-01-18